Finite covers of groups by cosets or subgroups

الأصل · EN

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let a₁G₁,...,aₖGₖ be left cosets in a group G such that aᵢGᵢᵢ₌₁ᵏ covers each element of G at least m times but none of its proper subsystems does. We show that if G is cyclic, or G is finite and G₁,...,Gₖ are normal Hall subgroups of G, then k≥ m+f([G:ᵢ₌₁ᵏGᵢ]), where f(∏ₜ₌₁ʳ pₜαᵗ)=∑ₜ₌₁ʳαₜ(pₜ-1) if p₁,...,pᵣ are distinct primes and α₁,...,αᵣ are nonnegative integers. When all the aᵢ are the identity element of G and all the Gᵢ are subnormal in G, we prove that there is a composition series from ᵢ₌₁ᵏGᵢ to G whose factors are of prime orders. The paper also includes some other results and two challenging conjectures.